How water retention in fractal soils depends on particle and pore sizes, shapes, volumes and surface areas

Water retention in soils as a function of suction is important in many disciplines, including engineering when assessing soil strength in infrastructure, and land management, agriculture and eco-hydrology. Water retention is described mathematically using a soil-water characteristic curve (SWCC), and many equations have been proposed which link degree of saturation, suction and voids ratio. They are empirical and phenomenological in origin, rarely incorporate both the particle size distribution and a description of pore geometry, and may not be valid for a soil in which pore surface area must remain constant and equal to particle surface area. Here, focusing on fractal soils, by setting particle and pore surface areas equal and constant, analytical derivations are presented linking all parameters defining SWCCs to particle and pore geometry information, size distributions, shapes, volumes and surface areas. Descriptions of how pore shapes and volumes depend on voids ratio are incorporated. The derivations show two key parameters, the air entry value and air expulsion value, are linked to the voids ratio in power laws, giving theoretical justification to what is observed in experiments. The power exponent is the fractal dimension of the particle size distribution. The voids ratio dependent SWCCs provide very good fits to data for six soils. This discovery means that a SWCC for a single voids ratio can be made applicable to any other voids ratio using just the particle size distribution. It is anticipated that extending these ideas to non-fractal soils may involve replacing the fractal size distributions with size distributions of other mathematical forms to capture size, shape, volume and surface area dependencies.